Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Generic Case

The objective of this article is to complete preliminary results from the work of Bonnard and Sugny (2009) and Sugny et al. (2007), concerning the time-minimal control of dissipative two-level quantum systems whose dynamics is governed by the Lindblad equation. The extremal system is described by a 3D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.

[1]  J. Gauthier,et al.  Optimal control in laser-induced population transfer for two- and three-level quantum systems , 2002 .

[2]  A. I. Solomon,et al.  Constraints on relaxation rates for N-level quantum systems , 2003, quant-ph/0312231.

[3]  J. Swoboda.  Time-optimal Control of Spin Systems , 2006, quant-ph/0601131.

[5]  B. Piccoli,et al.  Optimal Syntheses for Control Systems on 2-D Manifolds , 2004 .

[6]  U. Boscain,et al.  On the minimum time problem for driftless left-invariant control systems on SO(3) , 2002 .

[7]  Bernard Bonnard,et al.  Geodesic flow of the averaged controlled Kepler equation , 2009 .

[8]  Jean-Baptiste Caillau,et al.  Conjugate and cut loci of a two-sphere of revolution with application to optimal control , 2009 .

[9]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[10]  T. Seideman,et al.  Intense laser alignment in dissipative media as a route to solvent dynamics. , 2005, Physical review letters.

[11]  Bernard Bonnard,et al.  Time-Minimal Control of Dissipative Two-Level Quantum Systems: The Integrable Case , 2009, SIAM J. Control. Optim..

[12]  Constantin Carathéodory,et al.  Calculus of variations and partial differential equations of the first order , 1965 .

[13]  Dominique Sugny,et al.  Field‐free molecular alignment of CO2 mixtures in presence of collisional relaxation , 2008 .

[14]  U. Boscain,et al.  A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds , 2006, math/0609566.

[15]  Jean-Baptiste Caillau,et al.  Second order optimality conditions in the smooth case and applications in optimal control , 2007 .

[16]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[17]  David Bao,et al.  Zermelo navigation on Riemannian manifolds , 2003 .

[18]  Bernard Bonnard,et al.  UNE APPROCHE G EOM ETRIQUE DU CONTR ^ OLE OPTIMAL DE L'ARC ATMOSPH ERIQUE DE LA NAVETTE SPATIALE , 2002 .

[19]  S. Schirmer,et al.  Orbits of quantum states and geometry of Bloch vectors for N-level systems , 2003, quant-ph/0308004.

[20]  Heinz Schättler,et al.  The local structure of time-optimal trajectories in dimension three under generic conditions , 1988 .

[21]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[22]  U. Boscain,et al.  Time minimal trajectories for a spin 1∕2 particle in a magnetic field , 2005, quant-ph/0512074.

[23]  C. Altafini,et al.  QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC) 2357 Controllability properties for finite dimensional quantum Markovian master equations , 2002, quant-ph/0211194.

[24]  Jean-Paul Gauthier,et al.  On the K + P Problem for a Three-Level Quantum System: Optimality Implies Resonance , 2002 .

[25]  Bernard Bonnard,et al.  Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust , 2007 .

[26]  E B Lee,et al.  Foundations of optimal control theory , 1967 .

[27]  D. Sugny,et al.  Time-optimal control of a two-level dissipative quantum system , 2007, 0708.3794.

[28]  H. Jauslin,et al.  Control of Quantum Dynamics by Laser Pulses: Adiabatic Floquet Theory , 2003 .